The vector
in a modal representation (Eq.
(G.21)) specifies how
the modes are driven by the input. That is, the
th mode
receives the input signal
weighted by
. In a computational
model of a drum, for example,
may be changed corresponding to
different striking locations on the drumhead.

The vector
in a modal representation (Eq.
(G.21)) specifies how
the modes are to be mixed into the output. In other words,
specifies how the output signal is to be created as a
linear combination of the mode states:

In a computational model of an electric guitar string, for example,
changes whenever a different pick-up is switched in or
out (or is moved [99]).

The modal representation is not unique since
and
may be scaled in compensating ways to produce the same transfer
function. (The diagonal elements of
may also be permuted along
with
and
.) Each element of the state vector
holds the state of a single first-order mode of the system.

For oscillatory systems, the diagonalized state transition matrix must
contain complex elements. In particular, if mode
is both
oscillatory and undamped (lossless), then an excited
state-variable
will oscillate sinusoidally,
after the input becomes zero, at some frequency
, where

relates the system eigenvalue
to the oscillation frequency
, with
denoting the sampling interval in seconds.
More generally, in the damped case, we have

In practice, we often prefer to combine complex-conjugate pole-pairs
to form a real, ``block-diagonal'' system; in this case, the
transition matrix
is block-diagonal with two-by-two real matrices
along its diagonal of the form

where
is the pole radius, and
re
. Note that, for real systems, a real second
order block requires only two multiplies (one in the lossless case)
per time update, while a complex second-order system
requires two complex multiplies. The function
cdf2rdf() in the Matlab Control Toolbox
can be used to convert complex diagonal form to real block-diagonal
form.